Monotone Hurwitz Numbers and the HCIZ Integral

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BLAISE PASCAL

I. P. Goulden, Mathieu Guay-Paquet & Jonathan Novak Monotone Hurwitz Numbers and the HCIZ Integral

Volume 21, n^o1 (2014), p. 71-89.

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Monotone Hurwitz Numbers and the HCIZ Integral

I. P. Goulden Mathieu Guay-Paquet

Jonathan Novak

Abstract

In this article, we prove that the complex convergence of the HCIZ free energy is equivalent to the non-vanishing of the HCIZ integral in a neighbourhood ofz= 0.

Our approach is based on a combinatorial model for the Maclaurin coefficients of the HCIZ integral together with classical complex-analytic techniques.

Les nombres de Hurwitz monotones et l’intégrale HCIZ

Résumé

Nous démontrons que la convergence de l’énergie libre de l’intégrale HCIZ dans le plan complexe est équivalente à la non-nullité de l’intégrale HCIZ autour de z= 0. Notre approche est basée sur un modèle combinatoire pour les coefficients de Maclaurin de l’intégrale HCIZ et sur des méthodes classiques d’analyse complexe.

1. Introduction

The Harish-Chandra-Itzykson-Zuber integral, IN(z) =

Z

U(N)

e^zN^Tr(A^N^{U B}^N^U⁻¹⁾dU, (1.1) is a ubiquitous special function which plays a key role in random matrix theory and related areas. It enters into both the fine-scale spectral analysis of a single random matrix and the macroscopic analysis of several

The first author is partially supported by NSERC.

The second author is supported by NSERC.

Keywords:Matrix models, Hurwitz numbers, asymptotic analysis.

Math. classification:05E10, 15B62, 14N10.

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interacting random matrices. For further information on these and other aspects of the HCIZ integral, we refer the reader to the surveys [5, 11, 22].

In (1.1), the integration is over the group of N ×N unitary matrices against the normalized Haar measure, z is a complex parameter, and AN, BN are any two N ×N complex diagonal matrices. Since U(N) is compact, I_N(z) is an entire function of the complex variable z. Let Ω_N be a simply connected open set in Cwhich contains the origin and avoids the zeros of IN(z). The existence of such a domain is guaranteed by the discreteness of zeros of analytic functions together with the evaluation IN(0) = 1. Let logIN(z) denote the principal branch of the logarithm of I_N(z) on Ω_N. The holomorphic function

FN(z) = 1

N² logIN(z), z∈Ω_N,

is known in the random matrix literature as the free energy of the HCIZ integral.

A longstanding conjecture asserts that, if (A_N)^∞_N=1 and (B_N)^∞_N=1 are two sequences of complex diagonal matrices which grow in a suitably regular fashion, then F_N(z) converges uniformly on compact subsets of a complex neighbourhood of z= 0. This conjecture is implicit in the classic paper [13], and explicitly stated in [1, 2]. Clearly, a necessary condition for this conjecture to hold is the existence of a domain Ω containing the origin such that IN(z) is non-vanishing on Ω for all but finitely many N. In this paper, we will prove that this condition is also sufficient.

2. The Leading Derivatives Theorem

Let S(d) denote the symmetric group acting on {1, . . . , d}, and identify S(d) with its (right) Cayley graph as generated by the full conjugacy class of transpositions. Define an edge labelling of the Cayley graph as follows:

each edge corresponding to the transposition τ = (s t) is marked by t, the larger of the two numbers interchanged. Thus, emanating from each vertex of the Cayley graph, there is one 2-edge, two 3-edges, three 4-edges, etc.

Definition 2.1. A walk on the Cayley graph ofS(d) is said to bemono- tone if the labels of the edges it traverses form a weakly increasing sequence.

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Given two permutationsρ, σ∈S(d), we denote by w~^r(ρ, σ) the number ofr-step monotone walks fromρtoσ. Given two Young diagramsα, β `d, we denote by

W~ ^r(α, β) = ^X

ρ∈Cα

X

σ∈C_β

~

w^r(ρ, σ) (2.1)

the number of monotone walks beginning in the conjugacy class Cα and ending in the conjugacy class C_β.

Theorem 2.2 (Leading Derivatives Theorem). For anyN ≥1 and any 1≤d≤N, we have

I_N^(d)(0) =

∞

X

r=0

− 1 N

r

X

α,β`d

pα(AN)pβ(BN)W~ ^r(α, β), and this series is absolutely convergent. Equivalently, we have

IN(z) = 1 +

N

X

d=1

z^d d!

∞

X

r=0

− 1 N

r

X

α,β`d

pα(AN)pβ(BN)W~ ^r(α, β) +O(z^N⁺¹), where the O-term is uniform on compact subsets of C. Here

p_α(A_N) =

`(α)

Y

i=1

Tr(A^α_Nⁱ), p_β(B_N) =

`(β)

Y

j=1

Tr(B^β_N^j),

are the power sum symmetric functionspα, p_β evaluated on the eigenvalues of A_N, B_N.

In the remainder of this section, we give the proof of Theorem 2.2.

2.1. Differentiation under the integral sign

The derivatives of the entire function (1.1) may be computed by differentiation under the integral sign. In particular, the Maclaurin coefficients of IN(z) are given by

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I_N^(d)(0) =N^d Z

U(N)

(TrA_NU B_NU⁻¹)^ddU

=N^d^X

i,j

a_i(1). . . a_i(d)b_j(1). . . b_j(d) Z

U(N)

|u_i(1)j(1). . . u_i(d)j(d)|²dU,

where the summation is over all N^2d pairs of functions i, j:{1, . . . , d} → {1, . . . , N}, and A_N = diag(a₁, . . . , a_N),B_N = diag(b₁, . . . , b_N).

2.2. The Weingarten function

The integration of monomial functions of matrix elements overU(N) can be addressed using the Weingarten convolution formula of Collins and Śniady [3]:

Z

U(N)

u_i(1)j(1). . . u_i(d)j(d)u_i0(1)j⁰(1). . . u_i0(d)j⁰(d)dU = X

ρ,σ∈S(d)

δ_i,i⁰_ρδ_j,j⁰_σWg_N(ρ⁻¹σ),

where Wg_N :S(d)→Qis the Weingarten function, which is given by Wg_N(π) =

Z

U(N)

u₁₁. . . u_ddu_1π(1). . . u_dπ(d)dU

for N ≥d.

Given a function i:{1, . . . , d} → {1, . . . , N}, we denote by Stab(i) the set of permutations π ∈ S(d) such that iπ = i, and given a permuta- tion π ∈ S(d) we denote by Fix(π) the set of functions i : {1, . . . , d} → {1, . . . , N}such thatiπ=i. Applying the Weingarten convolution formula toI_N^(d)(0) with 1≤d≤N, we obtain

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I_N^(d)(0) =N^d^X

i

X

j

a_i(1). . . a_i(d)b_j(1). . . b_j(d) ^X

ρ∈Stab(i)

X

σ∈Stab(j)

Wg_N(ρ⁻¹σ)

=N^d^X

ρ

X

σ

Wg_N(ρ⁻¹σ) ^X

i∈Fix(ρ)

X

j∈Fix(σ)

a_i(1). . . a_i(d)b_j(1). . . b_j(d)

=N^d^X

ρ,σ

Wg_N(ρ⁻¹σ)p_t(ρ)(AN)p_t(σ)(BN).

(2.2) 2.3. 1/N-expansion of the Weingarten function

In [16] (see also [15]), it was shown that, for any N ≥ d and any π ∈ S(d), the Weingarten function admits the following absolutely convergent expansion in powers of 1/N:

Wg_N(π) = 1 N^d

∞

X

r=0

(−1)^rw~^r(id, π) N^r . Thus

Wg_N(ρ⁻¹σ) = 1 N^d

∞

X

r=0

(−1)^rw~^r(id, ρ⁻¹σ)

N^r = 1

N^d

∞

X

r=0

(−1)^rw~^r(ρ, σ) N^r . Plugging this expansion into (2.2) and changing order of summation, we arrive at

I_N^(d)(0) = ^X

ρ,σ∈S(d)

p_t(ρ)(A_N)p_t(σ)(B_N)

∞

X

r=0

(−1)^rw~^r(ρ, σ) N^r

=

∞

X

r=0

− 1 N

^r X

ρ,σ∈S(d)

p_t(ρ)(A_N)p_t(σ)(B_N)w~^r(ρ, σ),

where t(ρ) ` d is the cycle type of ρ, and likewise for t(σ). The internal sum may be written

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X

ρ,σ∈S(d)

p_t(σ)(AN)p_t(ρ)(BN)~w^r(ρ, σ) =X

α`d

X

β`d

pα(AN)pβ(BN) X

ρ∈Cα

X

σ∈Cβ

~ w^r(ρ, σ)

=X

α`d

X

β`d

pα(AN)pβ(BN)W~ ^r(α, β),

and this completes the proof of Theorem 2.2.

3. (Monotone) Hurwitz theory and the HCIZ free energy

3.1. Classical Hurwitz numbers

As shown by Hurwitz in the 19th century [12], the enumeration of unre- stricted walks on the symmetric group with given boundary conditions is equivalent to the enumeration of branched covers of the Riemann sphere with given singular data.

To state this precisely, consider the generating function W= 1 +

∞

X

d=1

z^d d!

∞

X

r=0

t^r r!

X

α,β`d

pα(A)pβ(B)W^r(α, β), where

A=





 a₁

a2

. ..





, B =





 b₁

b2

. ..







are a pair of formal infinite diagonal matrices and W^r(α, β) is the total number ofr-step walks onS(d) which begin inCαand end inCβ. ThusW is an element of the formal power series algebraQ[[z, t, a₁, a₂, . . . , b₁, b₂, . . .]].

Set

H^r(α, β) = z^d

d!

t^r

r!pα(A)pβ(B)

H,

where H= logW and [X]Y denotes the coefficient of termX in a series Y. By the exponential formula [9], the coefficientH^r(α, β) is the number of r-step walks beginning in C_α and ending in C_β whose endpoints and steps together generate a transitive subgroup of S(d).

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Hurwitz showed that H^r(α, β)/d! may be interpreted as a weighted count of degree dbranched covers of the Riemann sphere by a compact, connected Riemann surface such that the covering map has profileα over

∞, β over 0, and simple branching over each of the r^th roots of unity.

According to the Riemann-Hurwitz formula, such a cover exists if and only if

g= r+ 2−`(α)−`(β) 2

is a nonnegative integer, in which case it is the topological genus of the covering surface. We will use the notation H^r(α, β) = Hg(α, β), with the understanding that r and g determine one another via the Riemann- Hurwitz formula.

The numbersHg(α, β) were first considered from a modern perspective by Okounkov [17], who called them thedouble Hurwitz numbers. Proving a conjecture of Pandharipande [18] in Gromov-Witten theory, Okounkov showed that His a solution of the 2D Toda lattice hierarchy of Ueno and Takasaki. It was subsequently shown by Kazarian and Lando [14] that, when combined with the ELSV formula [4], Okounkov’s result implies the celebrated Kontsevich-Witten Theorem relating intersection theory in moduli spaces of Riemann surfaces to the KdV hierarchy.

3.2. Monotone Hurwitz numbers

Mimicking the above classical construction, consider the generating function

W~ = 1 +

∞

X

d=1

z^d d!

∞

X

r=0

t^r ^X

α,β`d

p_α(A_N)p_β(B_N)W~ ^r(α, β)

enumerating monotonewalks of all possible lengths and boundary conditions on all of the finite symmetric groups. Note that, due to the mono- tonicity constraint, the variable t is now an ordinary rather than exponential marker for the walk length statistic. Define the monotone double Hurwitz numbers by

H~^r(α, β) = z^d

d!t^rp_α(A)p_β(B)

H,~

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where H~ = logW. Then~ H~^r(α, β) is the number of r-step monotone walks beginning in C_α and ending in C_β whose endpoints and steps together generate a transitive subgroup of S(d). Alternatively, H~^r(α, β)/d!

counts a combinatorially restricted subset of the branched covers counted by H^r(α, β)/d!. In [7, 8], an extensive combinatorial theory of monotone Hurwitz numbers was developed, and monotone analogues of most combinatorial results in classical Hurwitz theory were obtained. Here we will use those results in tandem with Theorem 2.2.

3.3. The HCIZ free energy

The monotone double Hurwitz numbers describe the Maclaurin coefficients of the HCIZ free energy.

As in the Introduction, let Ω_N be a simply connected open set in C which contains the origin and avoids the zeros of I_N(z), and let logI_N(z) denote the principal branch of the logarithm of IN(z) on ΩN. From The- orem 2.2 and the exponential formula, it follows that the Maclaurin series of the logarithm is

logI_N(z) =

N

X

d=1

z^d d!

∞

X

r=0

− 1 N

^r X

α,β`d

p_α(A_N)p_β(B_N)H~^r(α, β) +O(z^N+1), where the O-term is uniform on compact subsets of ΩN.

The Maclaurin series of logI_N(z) can be simplified using the Riemann- Hurwitz formula. We have:

logIN(z) =

N

X

d=1

z^d d!

X

α,β`d

pα(AN)pβ(BN)

∞

X

r=0

− 1 N

r

H~^r(α, β) +O(z^N⁺¹)

=

N

X

d=1

z^d d!

X

α,β`d

pα(AN)pβ(BN)

∞

X

g=0

−1 N

2g−2+`(α)+`(β)

H~g(α, β) +O(z^N⁺¹)

=N²

N

X

d=1

z^d d!

∞

X

g=0

1 N^2g

X

α,β`d

(−1)^`(α)+`(β)p_α(A_N) N^`(α)

p_β(B_N) N^`(β)

H~_g(α, β) +O(z^N⁺¹),

with theO-term uniform on compact subsets of Ω_N. Thus, parameterizing the monotone double Hurwitz numbers by g instead of r, we arrive at a

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description of F_N(z) =N⁻²logI_N(z) which is well-poised for anN → ∞ asymptotic analysis.

3.4. Asymptotic expansion of Maclaurin coefficients

We will now consider the asymptotic behaviour of the Maclaurin coefficients of F_N(z) as N → ∞, when A_N, B_N vary regularly with N.

Suppose there exists a nonnegative number M, a nonnegative integer h, and two complex sequencesφ_k, ψ_k such that:

(1) kA_Nk,kB_Nk ≤M for all N ≥1;

(2) For eachk≥1, 1

N Tr(A^k_N) =φ_k+o 1

N^2h

, 1

N Tr(B_N^k) =ψ_k+o 1

N^2h

asN → ∞.

We will summarize these conditions by saying that AN, BN are (M, h)- regular with limit moments φ_k, ψ_k.

Theorem 3.1. Suppose that A_N, B_N are (M, h)-regular with limit mo- ments φ_k, ψ_k. Then, for eachd≥1, we have

F_N^(d)(0) =

h

X

g=0

C_g,d N^2g +o

1 N^2h

as N → ∞, where

C_g,d= ^X

α,β`d

(−1)^`(α)+`(β)φ_αψ_βH~_g(α, β), and

φ_α=

`(α)

Y

i=1

φ_α_i, ψ_β =

`(β)

Y

j=1

φ_β_j.

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Proof. For 1≤d≤N, we have F_N^(d)(0) =

∞

X

g=0

C_g,d,N N^2g ,

where

Cg,d,N = ^X

α,β`d

(−1)^`(α)+`(β)pα(AN) N^`(α)

p_β(B_N) N^`(β)

H~g(α, β).

Since kA_Nk,kB_Nk ≤M for all N ≥1, we have

|C_g,d,N| ≤M^2d ^X

α,β`d

H~g(α, β)

for all N ≥1. Since H~g(α, β) counts certain solutions of the equation σ =ρτ1. . . τr

in S(d), with r= 2g−2 +`(α) +`(β), we have H~_g(α, β)≤(d!)^2g+2d for all α, β `d. Consequently,

|C_g,d,N| ≤(d!p(d)M)^2d(d!)^2g,

wherep(d) is the number of Young diagrams with dcells. Moreover, since AN, BN areh-regular with limit momentsφ_k, ψ_k, we have

C_g,d,N =C_g,d+o 1

N^2h

as N → ∞. We thus have

F_N^(d)(0) =

h

X

g=0

C_g,d,N N^2g +

∞

X

g=h+1

C_g,d,N N^2g

=

h

X

g=0

C_g,d N^2g +o

1 N^2h

+O

1 N^2h+2

as N → ∞, which proves the claim.

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Remark 3.2. The convergence of F_N^(d)(0) under the above hypotheses was first stated by Itzykson and Zuber [13], and proved by Collins [1]. Collins obtained the limit ofF_N^(d)(0) as a double sum overS(d), whereas we present the same limit as a double sum over partitions of d.

Remark 3.3. To the best of our knowledge, Theorem 3.1 is the first result which addresses the sub-leading asymptotics of F_N^(d)(0), clearly showing the emergence of a “topological expansion” in this context.

4. Absolute summability of genus-specific generating func- tions

Let us introduce a sequence of formal power series Cg(z)∈C[[z]] defined by

Cg(z) =

∞

X

d=1

z^d

d!Cg,d, g≥0, (4.1)

where C_g,d are the expansion coefficients from Theorem 3.1. The goal of this section is to prove that these formal power series are absolutely summable, and to obtain bounds on their radii of convergence. By the uniform boundedness ofkA_Nkand kB_Nk, this reduces to establishing the absolute summability of the series

H~_g(z) =

∞

X

d=1

z^d d!

X

α,β`d

H~_g(α, β), g≥0. (4.2)

4.1. Monotone simple Hurwitz numbers

Consider the genus-specific generating functions

~S_g =

∞

X

d=1

z^d d!

H~_g,d for the monotone simple Hurwitz numbers

H~g,d=H~g(1^d,1^d).

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The monotone simple Hurwitz number H~_g,d counts monotone loops of length r = 2g−2 + 2d, based at any given point of S(d), whose steps generate a transitive subgroup of S(d).

According to [8, Theorem 1.4], for any g≥2 we have

~S_g = ζ(1−2g)

2−2g + 1

(1−6s)^2g−2

3g−3

X

r=0

X

µ`r

cg,µ(6s)^`(µ) (1−6s)^`(µ),

whereζ is the Riemann zeta function, thec_g,µ’s are rational numbers, and s is the unique solution of the functional equation

s=z(1−2s)⁻²

in the formal power series algebra Q[[z]]. This equation may be solved by Lagrange inversion, yielding the solution

s=

∞

X

n=1

2ⁿ⁻¹ n

3n−2 n−1

! zⁿ.

Since

s⁰ =

∞

X

n=0

3n+ 1 n

!

(2z)ⁿ=₂F₁ 2

3,4 3,3

2;27 2 z

,

where ₂F₁(a, b, c;z) is the Gauss hypergeometric function, ~S_g extends to a holomorphic function of zon the domainC\[z_c,∞), where

zc= 2 27.

In the case g= 0, [7, Theorem 1.1] yields the exact formula H~_0,d

d! = 2^d−1 d²(d−1)

3d−3 d−1

! ,

so that, using Stirling’s formula, we obtain ₂₇² as the radius of convergence of ~S₀. Moreover, since~S₀ has positive coefficients, Pringsheim’s Theorem (see [20, §7.21]) guarantees that zc= ₂₇² is a singularity of ~S₀.

We have thus shown that:

Theorem 4.1. The series~S_g,g≥0, have a common dominant singularity at the critical point zc= ₂₇² .

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Remark 4.2. The g= 0 case of Theorem 4.1 was obtained by Zinn-Justin in [21] using the Toda lattice equations. For a combinatorial solution of the Toda equations encompassing those of Okounkov and Zinn-Justin, see [6].

Remark 4.3. P. Di Francesco has pointed out to us that the critical value zc= ₂₇² also appears in the enumeration of finite groups, see [19].

4.2. Monotone double Hurwitz numbers

Consider the full genus-specific generating functions H~_g(z) defined in equation (4.2). Obviously,

X

α,β`d

H~_g(α, β)≥H~_g,d,

so the radius of convergence ofH~_g is at most the radius of convergence of

~S_g.

We now explain how a lower bound on the radius of convergence of H~_g follows from the study of a refinement of the monotone double Hurwitz number H~g(α, β). Given a positive integer c, let H~g(α, β;c) denote the number of walks counted by H~g(α, β) whose steps have c distinct labels.

The following inequality is obtained in [10]:

d

X

c=2

3^cH~g(1^d,1^d;c)≤ ^X

α,β`d

H~g(α, β)≤

d

X

c=2

4^cH~g(1^d,1^d;c).

The proof is combinatorial, and makes use of an action of the symmetric group S(r) on the set of walks counted by the classical double Hurwitz number H_g(α, β). From this inequality and the definition of H~_g(α, β;c), we obtain

X

α,β`d

H~g(α, β)≤4^d−1

d

X

c=2

H~g(1^d,1^d;c) = 4^d−1H~_g,d,

which implies that the radius of convergence of H~_g is at least one quarter the radius of convergence of ~S_g. Combining this with Theorem 4.1, we have:

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Theorem 4.4. For eachg≥0, the seriesH~_g is absolutely summable, and has radius of convergence at least ₅₄¹ and at most ₂₇².

Remark 4.5. The results of [2] also imply the absolute summability ofH~₀, but without effective bounds on the radius of convergence.

5. Convergence of the HCIZ free energy

5.1. Convergence under a non-vanishing hypothesis

Let AN, BN be (M,0)-regular with limit moments φ_k, ψ_k, respectively.

Then, by Theorem 4.4, the series C₀(z) is absolutely summable, with radius of convergence at least _54M¹ ₂.

Theorem 5.1. Suppose that there exists a positive number R such that IN(z) is non-vanishing on the open discD(0, R) for all but finitely many N. Let

r = min

R, 1 54M²

.

Then F_N(z)→C₀(z) uniformly on compact subsets of D(0, r).

Proof. First, we show that the claim holds pointwise. The proof is a two circles argument. Let z0 ∈ D(0, r) and ε > 0 be arbitrary. Choose r1, r2

such that

|z₀|< r₁ < r₂ < r.

For N sufficiently large, we have

|F_N(z₀)−C₀(z₀)| ≤

∞

X

d=1

|F_N^(d)(0)−C_0,d||z₀|^d d! . By Cauchy’s inequality,

1

d!|F_N^(d)(0)−C0,d| ≤ kF_N −C0k_r₁ r₁^d ,

where k · k_r₁ denotes sup-norm over the circle of radius r1. We thus have

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|F_N(z0)−C₀(z0)| ≤

E

X

d=1

|F_N^(d)(0)−C_0,d||z₀|^d

d! +kF_Nk_r₁+kC₀k_r₁ 1−^|z_r⁰^|

1

|z₀| r₁

E+1

for any positive integer E.

We will now bound the error term independently of N. Since

|I_N(z)| ≤ Z

U(N)

e^|z|N|^Tr(A^N^{U B}^N^U⁻¹^)|dU

≤ Z

U(N)

e^|z|M²^N PN

i,j=1|u_ij|²

dU

≤e^M²^N²^|z|

for all z ∈ C, where the last line follows from the fact that (|u_ij|²) is a doubly stochastic matrix, the inequality

<F_N(z)≤M²|z|

holds for allz∈Ω_N, the domain of holomorphy ofFN(z). Combining this with the Borel-Carathéodory inequality, which bounds the sup-norm of an analytic function on a circle in terms of the maximum of its real part on a circle of larger radius (see e.g. [20, §5.5]), we have

kF_Nk_r₁ ≤ 2r1

r2−r1

sup

|z|=r2

<F_N(z)≤ 2M²r1r2

r2−r1

.

Returning to our estimate on |F_N(z0)−C0(z0)|, we now have the inequality

|F_N(z₀)−C₀(z₀)| ≤

E

X

d=1

|F_N^(d)(0)−C_0,d||z₀|^d d! +

2M²r1r2

r2−r1 +kC₀k_r₁ 1−^|z_r⁰^|

1

|z₀| r₁

E+1

for all N sufficiently large and all E ≥ 1. Choosing E0 large enough so that

2M²r1r2

r2−r1 +kC₀k_r₁ 1−^|z_r⁰^|

1

|z₀| r1

^E+1

< ε 2,

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and subsequently choosing N0 large enough so that

E0

X

d=1

|F_N^(d)

0(0)−C0,d||z₀|^d d! < ε

2, we obtain that

|F_N(z₀)−C₀(z₀)|< ε for all N ≥N0. This completes the proof.

We now explain how the mode of convergence can be boosted from pointwise to uniform on compact subsets of D(0, r). It is easy to check that

sup

N

kF_Nk_K <∞

for each compact set K⊂D(0, r); for example, one could prove this holds for closed discs by using the Borel-Carathéodory inequality again. Thus {F_N} is a locally uniformly bounded family on D(0, r), and Vitali’s the- orem [20, §5.21] implies pointwise and uniform-on-compact convergence

are equivalent.

5.2. Remarks on the non-vanishing hypothesis

Theorem 5.1 reduces the complex convergence of FN(z) near z = 0 to checking that the zeros of I_N(z) do not encroach on z = 0 as N → ∞.

Therefore it is of significant interest to determine sufficient hypotheses on the matrix sequencesA_N andB_N which ensure that this condition holds.

Determining the exact locations of the zeros of I_N(z) seems to be a difficult problem in general. Let us give one example where it can be solved explicitly. Suppose that A_N andB_N are real diagonal with distinct eigenvalues

a₁ <· · ·< a_N, b₁ <· · ·< b_N,

and suppose further that the eigenvalues of AN form an arithmetic pro- gression with constant difference ~> 0. In this case, all determinants in the HCIZ formula (see [13, Equation 3.28]) are Vandermonde determinants, and one computes that the zeros of I_N(z) are the pure imaginary points

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z= 2kπ

N~(b_j−b_i)i, 1≤i < j ≤N, k∈Z.

Thus one has a sort of Lee-Yang theorem in this special case. If the eigenvalues of A_N and B_N are confined to a fixed interval [−M, M] for all N, then ~ is of order 1/N and the above formula shows that the zeros of I_N(z) remain at bounded distance from the origin as N increases.

Concerning the verification of the non-vanishing hypothesis more gener- ally, we do not know much at present. It might be that the Toda equations could be of help here, but we have not pursued this seriously.

5.3. A conjecture

In this article, we have produced explicit holomorphic candidatesC_g(z) for the asymptotics of F_N(z) near z= 0. Indeed, ifF_N(z) admits a uniform- on-compact asymptotic expansion on the asymptotic scale N⁻² in some neighbourhood Ω of z= 0, then the coefficients in this expansionmust be the functions Cg(z). We therefore believe that the following conjecture is reasonable: IfA_N, B_N are (M, h)-regular with limit momentsφ_k, ψ_k, then there exists a neighbourhood Ω of z= 0 such that

FN(z) =

h

X

g=0

Cg(z) N^2g +o

1 N^2h

uniformly on compact subsets of Ω.

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I. P. Goulden

Department of Combinatorics &

Optimization

University of Waterloo 200 University Avenue West Waterloo, ON N2L 3G1 Canada

[email protected]

Mathieu Guay-Paquet LaCIM

Université du Québec à Montréal 201 Avenue du Président-Kennedy Montréal, QC H2X 3Y7

Canada

[email protected]

Jonathan Novak

Department of Mathematics

Massachusetts Institute of Technology 77 Massachusetts Ave.

Boston, MA 02114 USA

[email protected]